MTH 233 STATISTICSDownload Link: https://uopcourses.com/category/mth-233/ 1 Which of the following statistics are unbiased estimators of population parameters? Choose the correct answer below. Select all that apply. A. Sample median used to estimate a population median. B. Sample proportion used to estimate a population proportion. C. Sample standard deviation used to estimate a population standard deviation. D. Sample range used to estimate a population range. E. Sample variance used to estimate a population variance. F. Sample mean used to estimate a population mean. 2 When two births are randomly selected, the sample space for genders is bb, bg, gb, and gg. Assume that those four outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from two births. Does the mean of the sample proportions equal the proportion of girls in two births? Does the result suggest that a sample proportion is an unbiased estimator of a population proportion? For the entire population, assume the probability of having a boy is one half12, the probability of having a girl is one half12, and this is not affected by how many boys or girls have previously been born. Determine the probabilities of each sample proportion. Sample proportion of girls 0 0.5 1 Does the mean of the sample proportions equal the proportion of girls in two births? A. Yes, both the mean of the sample proportions and the population proportion are one third13. B. Yes, both the mean of the sample proportions and the population proportion are one half12. C. Yes, both the mean of the sample proportions and the population proportion are one fourth14. D. No, the mean of the sample proportions and the population proportion are not equal. Does the result suggest that a sample proportion is an unbiased estimator of a population proportion? 3 Assume a population of 44, 55, and 99. Assume that samples of size nequals=2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below. 44,44 44,55 44,99 55,44 a. Find the value of the population standard deviation sigmaσ. sigmaσequals= (Round to three decimal places as needed.) b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability distribution of the distinct standard deviation values. Use ascending order of the sample standard deviations. s Probability 0 0.707 2.828 3.536 (Type integers or fractions.) c. Find the mean of the sampling distribution of the sample standard deviations. sigma overbarσequals= (Round to three decimal places as needed.) d. Do the sample standard deviations target the value of the population standard deviation? In general, do sample standard deviations make good estimators of population standard deviations? Why or why not? A. 55,55 The sample standard deviations do target the population standard deviation, therefore, sample standard deviations are unbiased estimators. B. The sample standard deviations do not target the population standard deviation, therefore, sample standard deviations are unbiased estimators. 4 The population of current statistics students has ages with mean muμ and standard deviation sigmaσ. Samples of statistics students are randomly selected so that there are exactly 6060 students in each sample. For each sample, the mean age is computed. What does the central limit theorem tell us about the distribution of those mean ages? Choose the correct answer below. A. Because ngreater than>30, the sampling distribution of the mean ages can be approximated by a normal distribution with mean muμ and standard deviation sigmaσ. B. Because ngreater than>30, the sampling distribution of the mean ages can be approximated by a normal distribution with mean muμ and standard deviation StartFraction sigma Over StartRoot 60 EndRoot EndFraction σ60. C. Because ngreater than>30, the sampling distribution of the mean ages is precisely a normal distribution with mean muμ and standard deviation StartFraction sigma Over StartRoot 60 EndRoot EndFraction σ60. D. Because ngreater than>30, the central limit theorem does not apply in this situation. 5 The capacity of an elevator is 1212 people or 20402040 pounds. The capacity will be exceeded if 1212 people have weights with a mean greater than 2040 divided by 12 equals 170 pounds.2040/12=170 pounds. Suppose the people have weights that are normally distributed with a mean of 179 lb179 lb and a standard deviation of 32 lb32 lb. (a) Find the probability that if a person is randomly selected, his weight will be greater than 170170 pounds. The probability is approximately (Round to four decimal places as needed.) (b) Find the probability that 1212 randomly selected people will have a mean that is greater than 170170 pounds. The probability is approximately (Round to four decimal places as needed.) (c) Does the elevator appear to have the correct weight limit? Why or why not? A. No, 12 randomly selected people will never be under the weight limit. B. No, there is a good chance that 12 randomly selected people will exceed the elevator capacity. C. Yes, there is a good chance that 12 randomly selected people will not exceed the elevator capacity. D. Yes, 12 randomly selected people will always be under the weight limit. 6 Which of the following is NOT a conclusion of the Central Limit Theorem? Choose the correct answer below. A. The standard deviation of all sample means is the population standard deviation divided by the square root of the sample size. B. The distribution of the sample means x overbarx will, as the sample size increases, approach a normal distribution. C. The mean of all sample means is the population mean muμ. D. The distribution of the sample data will approach a normal distribution as the sample size increases. 7 Fill in the blank. The _______ states that if, under a given assumption, the probability of a particular observed event is exceptionally small (such as less than 0.05), we conclude that the assumption is probably not correct. The states that if, under a given assumption, the probability of a particular observed event is exceptionally small (such as less than 0.05), we conclude that the assumption is probably not correct. 8 Find the critical value z Subscript alpha divided by 2zα/2 that corresponds to the given confidence level. 8383% z Subscript alpha divided by 2zα/2equals= (Round to two decimal places as needed.) 9 Express the confidence interval left parenthesis 0.049 comma 0.125 right parenthesis(0.049,0.125) in the form of ModifyingAbove p with caret minus Upper E less than p less than ModifyingAbove p with caret plus Upper E<="" path="">p−E<p<<="" path="">p+E. less than p less than<p< (Type integers or decimals.) 10 Use the sample data and confidence level given below to complete parts (a) through (d). A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, n equals 997n=997 and x equals 505x=505 who said "yes." Use a 90 %90% confidence level. LOADING... Click the icon to view a table of z scores. a) Find the best point estimate of the population proportion p. (Round to three decimal places as needed.) b) Identify the value of the margin of error E. Eequals= (Round to four decimal places as needed.) c) Construct the confidence interval. less than p less than<p< (Round to three decimal places as needed.) d) Write a statement that correctly interprets the confidence interval. Choose the correct answer below. A. There is a 9090% chance that the true value of the population proportion will fall between the lower bound and the upper bound. B. One has 9090% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion. C. 9090% of sample proportions will fall between the lower bound and the upper bound. 11 A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 670670 babies were born, and 335335 of them were girls. Use the sample data to construct a 9999% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective? less than< pless than< (Round to three decimal places as needed.) Does the method appear to be effective? NoNo, the proportion of girls is notis not significantly different from 0.5. This is the correct answer. YesYes, the proportion of girls isis significantly different from 0.5. 12 An online site presented this question, 'Would the recent norovirus outbreak deter you from taking a cruise?' Among the 34 comma 94734,947 people who responded, 6666% answered 'yes'. Use the sample data to construct a 9595% confidence interval estimate for the proportion of the population of all people who would respond 'yes' to that question. Does the confidence interval provide a good estimate of the population proportion? less than<p p less than< (Round to three decimal places as needed.) Does the confidence interval provide a good estimate of the population proportion? A. No, the responses are not independent. B. Yes, the sample is large enough to provide a good estimate of the population proportion. C. Yes, all the assumptions for a confidence interval are satisfied. D. No, the sample is a voluntary sample and might not be representative of the population. 13 In a poll of 517517 human resource professionals, 43.143.1% said that body piercings and tattoos were big grooming red flags. Complete parts (a) through (d) below. a) Among the 517517 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big grooming red flags? (Round to the nearest integer as needed.) b) Construct a 99% confidence interval estimate of the proportion of all human resource professionals believing that body piercings and tattoos are big grooming red flags. less than p less than<p< (Round to three decimal places as needed.) c) Repeat part (b) using a confidence level of 80%. less than p less than<p< (Round to three decimal places as needed.) d) Compare the confidence intervals from parts (b) and (c) and identify the interval that is wider. Why is it wider? Select the correct choice below and fill in the answer boxes to complete your choice. A. The confidence interval is wider than the confidence interval. As the confidence interval widens, the probability that the confidence interval actually does contain the population parameter increases. B. The confidence interval is wider than the confidence interval. As the confidence interval narrows, the probability that the confidence interval actually does contain the population parameter increases. C. The confidence interval is wider than the confidence interval. As the confidence interval widens, the probability that the confidence interval actually does contain the sample parameter increases. D. The confidence interval is wider than the confidence interval. As the confidence interval narrows, the probability that the confidence interval actually does contain the sample parameter increases. 14 A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.040.04 with 9595% confidence if a) she uses a previous estimate of 0.360.36? b) she does not use any prior estimates? a) nequals= (Round up to the nearest integer.) b) nequals= (Round up to the nearest integer.) 15 Do one of the following, as appropriate. (a) Find the critical value z Subscript alpha divided by 2zα/2, (b) find the critical value t Subscript alpha divided by 2tα/2, (c) state that neither the normal nor the t distribution applies. Confidence level 9898%; nequals=1919; sigma is knownσ is known; population appears to be very skewedvery skewed. Click here to view a table of critical t-values. LOADING... Click here to view page 1 of the standard normal table. LOADING... Click here to view page 2 of the standard normal table. LOADING... Find the critical value. A. t Subscript alpha divided by 2tα/2equals=2.2142.214 B. t Subscript alpha divided by 2 Baseline equals 2.552tα/2=2.552 C. z Subscript alpha divided by 2 Baseline equals 2.33zα/2=2.33 D. z Subscript alpha divided by 2zα/2equals=2.0552.055 E. Neither normal nor t distribution applies.Neither normal nor t distribution applies. 16 Do one of the following, as appropriate. (a) Find the critical value z Subscript alpha divided by 2zα/2, (b) find the critical value t Subscript alpha divided by 2tα/2, (c) state that neither the normal nor the t distribution applies. Confidence level 9595%; nequals=1717; sigma equals 21.1σ=21.1; population appears to be normally distributednormally distributed. Click here to view a table of critical t-values. LOADING... Click here to view page 1 of the standard normal table. LOADING... Click here to view page 2 of the standard normal table. LOADING... Find the critical value. A. t Subscript alpha divided by 2 Baseline equals 2.120tα/2=2.120 B. z Subscript alpha divided by 2 Baseline equals 1.96zα/2=1.96 C. z Subscript alpha divided by 2zα/2equals=1.6451.645 D. t Subscript alpha divided by 2tα/2equals=1.7461.746 E. Neither normal nor t distribution applies.Neither normal nor t distribution applies. 17 A data set includes 106106 body temperatures of healthy adult humans for which x overbarxequals=98.998.9degrees°F and s equals 0.62 degrees Upper Fs=0.62°F. Complete parts (a) and (b) below. Click here to view a t distribution table. LOADING... Click here to view page 1 of the standard normal distribution table. LOADING... Click here to view page 2 of the standard normal distribution table. LOADING... a. What is the best point estimate of the mean body temperature of all healthy humans? The best point estimate is degrees°F. (Type an integer or a decimal.) b. Using the sample statistics, construct a 9999% confidence interval estimate of the mean body temperature of all healthy humans. Do the confidence interval limits contain 98.6degrees°F? What does the sample suggest about the use of 98.6degrees°F as the mean bodytemperature? What is the confidence interval estimate of the population mean muμ? degrees°Fless than<muμless than<degrees°F (Round to three decimal places as needed.) Do the confidence interval limits contain 98.6degrees°F? YesYes NoNo What does this suggest about the use of 98.6degrees°F as the mean body temperature? A. This suggests that the mean body temperature could very possibly bevery possibly be 98.6degrees°F. B. This suggests that the mean body temperature could be higher thanbe higher than 98.6degrees°F. C. This suggests that the mean body temperature could be lower thanbe lower than 98.6degrees°F. 18 Twelve different video games showing substance use were observed and the duration times of game play (in seconds) are listed below. The design of the study justifies the assumption that the sample can be treated as a simple random sample. Use the data to construct a 9999% confidence interval estimate of muμ, the mean duration of game play. 40574057 38833883 38493849 40284028 43164316 Click here to view a t distribution table. LOADING... Click here to view page 1 of the standard normal distribution table. LOADING... Click here to view page 2 of the standard normal distribution table. LOADING... What is the confidence interval estimate of the population mean muμ? less than<muμless than< (Round to one decimal place as needed.) 19 An IQ test is designed so that the mean is 100 and the standard deviation is 1212 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 9595% confidence that the sample mean is within 22 IQ points of the true mean. Assume that sigmaσequals=1212 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation. The required sample size is (Round up to the nearest integer.) Would it be reasonable to sample this number of students? YesYes. This number of IQ test scores is a fairly smallsmall number. 48204820 NoNo. This number of IQ test scores is a fairly largelarge number. NoNo. This number of IQ test scores is a fairly smallsmall number. YesYes. This number of IQ test scores is a fairly largelarge number. 20 Fill in the blank. The _____________ is the best point estimate of the population mean. The is the best point estimate of the population mean.